# Full text of "Treatise On Analysis Vol-Ii"

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```1    EXISTENCE AND UNIQUENESS OF HAAR MEASURE       243

the formulas

(14.1.1)              (Y(4/X*) =f(s-*x)9       (8(5)/)(jc) =/(jcs).

It follows immediately from this definition that we have

(14.1 .1 .1 )             y(*0/= YO)(Y(0/),       S(*0/

for all j, t e G.

If /j is a (complex) measure on G, we denote by y(s)n and \$C?)ju the meas-
ures on G which are the images of fi under the homeomorphisms x\-^sx and
1, respectively (13.1.6), so that

(14.1.2)

for all functions /e Jfc(G). From this definition it follows that

(14.1 .2.1 )

for all s, t e G.

The measure /i is said to be left (resp. right) invariant if

(1 4.1 .2.2)                      y(s)ju = /*    (resp. 8(s)n = /x)

for all s e G.

If a measure jj. ^ 0 on G is left invariant, then Supp(/x) = G, because
Supp(j(s)fj) = ^ • Supp(|i) for all s e G by virtue of (1 3.1 9.4), and Supp(ju) ^ 0.
Similarly for right-invariant measures.

Let ^ be a left-invariant measure on G and let/be a //-integrable mapping
of G into R or C. Then for each ,seG the function xt-+f(s~lx) is also
/i-integrable, and we have

(14.1.2.3)

by (13.7.10). In particular, if A is a ju-integrable set, then so is ^A, and
(14.1.2.4)                                /x(jA) = ^(A).

If /is any mapping of G into a set E, we write
(14.1.3)                      /(*) ^/(jt-1)       for all   xeG. EXISTENCE AND UNIQUENESS OF HAAR MEASURE
```